It would appear that you have covered all of the possibilities. Based on your statements, it really comes down to the blend doors opening all the way or a partially blocked Heather core.
There's a easy check for the core, after the car is warmed up, feel both hoses going into the cabin. Both should be hot. In one is noticeably cooler than the other, there's a partial blockage.
Some time ago, Ford issued a customer satisfaction program for this generation due to casting sand or scale corroding and plugging the heater core channels. (It's expired now, but the problems you wrote about seem to describe it to a tee. Some sand left over from engine casting worked its way to the core partially blocking the small channels. Weaker on drivers side than passenger. Never gets really hot, etc.)
The complaint was for a 2008 but applies to all first generation edges.
https://blueovalforums.com/forums/index.php?/topic/37362-poor-heater-performance/